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In this paper we consider two piecewise Riemannian metrics defined on the Culler–Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil–Petersson metric on the Teichmüller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil–Petersson metric, from the point of view of geometric group theory, these metrics behave very differently than the Weil–Petersson metric. Specifically, we show that when the rank r is at least 4, the action of
$\operatorname {\mathrm {Out}}(\mathbb {F}_r)$
on the completion of the Culler–Vogtmann outer space using the entropy metric has a fixed point. A similar statement also holds for the pressure metric.
Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.
We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.
This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.
We classify all possible JSJ decompositions of doubles of free groups of rank two, and we also compute the Makanin–Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup separable.
We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We also prove this fact.
Given a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let ${{M}_{cb}}A(G)$ denote the completely bounded multipliers of $A(G)$, and let ${{A}_{Mcb}}\,(G)$ stand for the closure of $A(G)$ in ${{M}_{cb}}A(G)$. We characterize the norm one idempotents in ${{M}_{cb}}A(G)$: the indicator function of a set $E\,\subset \,G$ is a norm one idempotent in ${{M}_{cb}}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of ${{A}_{Mcb}}\,(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which ${{A}_{Mcb}}\,(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)
We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left-ordering of the free group on k generators, k>1, that has no convex subgroups.
Let $S$ be a subset of an amenable group $G$ such that $e\,\in \,S$ and ${{S}^{-1}}\,=\,S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented.
We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one of P. Brinkmann that one can determine whether two cyclic words in a free group are mapped to each other by some power of a given automorphism. We also solve the power conjugacy problem, and give an algorithm to recognize whether two given elements of a finitely generated free group are twisted conjugated to each other with respect to a given automorphism.
A group $G$ is said to have the Bergman property (the property of uniformity of finite width) if given any generating $X$ with $X=X^{-1}$ of $G$, we have that $G=X^k$ for some natural $k$, that is, every element of $G$ is a product of at most $k$ elements of $X$. We prove that the automorphism group $\operatorname{Aut}(N)$ of any infinitely generated free nilpotent group $N$ has the Bergman property. Also, we obtain a partial answer to a question posed by Bergman by establishing that the automorphism group of a free group of countably infinite rank is a group of uniformly finite width.
Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable $\aleph _{1}$-free group be a Polish group? A negative answer is given here; indeed, it is proved that an $\aleph _{1}$-free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non-commutative analogue of the notion of separable group.
Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.
We define a class of equations that are not amenable but are type K and are therefore solvable over torsion-free groups. Moreover, we show that these new equations are solvable over all groups.
We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.
An element of a free group F is called almost primitive in F, if it is primitive in every proper subgroup containing it, though not in F itself. Several examples of almost primitive elements (APEs) are exhibited. The main results concern the behaviour of proper powers wℓ of certain APEs w in a free group F (and, more generally, in free products of cycles) with respect to any subgroup H containing such a power “minimally“: these assert, in essence, that either such powers of w behave in H as do powers of primitives of F, or, if not, then they “almost” do so and furthermore H must then have finite index in F precisely determined by the smallest positive powers of conjugates of w lying in H. Finally, these results are applied to show that the groups of a certain class (potentially larger than that of finitely generated Fuchsian groups) have the property that all their subgroups of infinité index are free products of cyclic groups.
Let G be a finite group. A natural invariant c(G) of G has been defined by W.J. Ralph, as the order (possibly infinite) of a distinguished element of a certain abelian group associated to G. Ralph has shown that c(Zn) = 1 and c(Z2 ⴲ Z2) = 2. In the present paper we show that c(G) is finite whenever G is a dihedral group or a 2-group, and obtain upper bounds for c(G) in these cases.
We show that, if [s,t][u, v] = x2 in a free group, x need not be a commutator. We arrive at our example by use of a result of D. Piollet which characterizes solutions of such equations using an algebraic interpretation of the mapping class group of the corresponding surface.